Effect assessment for the interaction between shaking table and eccentric load

Electro-hydraulic shaking table is an essential experimental apparatus to evaluate structural performance under actual vibration condition. The control-structure interaction (CSI) between shaking table and eccentric load has lately received considerable attention for causing the accuracy degradation of shaking table test. At present, the research gap of the influence of the eccentricity of load on the CSI makes it challenging to find the CSI effects. And an effect assessment is yet to be proposed to evaluate the CSI effects, which has impeded the development of test technology. To overcome those theoretical bottlenecks, in this research, an analytical transfer function matrix of shaking table and eccentric load is established to analyze the CSI effects. The analysis is conducted under such conditions as different mass ratio (MR), moment of inertia ratio (IR), and eccentric distance ratio (ER) conditions. Through the analysis, the role of the ER is identified, the sensitivities of the MR, IR, and ER to the transfer function matrix are revealed, and the CSI effects are found. Furthermore, a novelty effect assessment is proposed to appraise whether the CSI effects can be ignored in shaking table test. And the visualization expression of the effect assessment is obtained for convenient application.


Scientific Reports
| (2022) 12:15349 | https://doi.org/10.1038/s41598-022-19743-y www.nature.com/scientificreports/ of structure, and the resonance peak affects the control performance of shaking table. Tang et al. 23 analyzed the CSI effects between shaking table and multi degree of freedom structure. The analysis revealed that the CSI exert effects on the MST at the previous natural frequency of structure. Guo et al. 24 analyzed the CSI effects on shaking table tests. The analysis showed that the nonlinearity of test structure has great impact on the MST and the control accuracy of the shaking table decreases. From the aspects of shaking table characteristics, many scholars have analyzed the effects of control parameters, the time delay and the bias of servo valve, the stiffness of hinge, the mass of hydraulic cylinder, the deformation of platform, and the coupling between two shaking tables on the MST (Fig. 1e). Trombetti et al. 20 studied the sensitivities of control gain (PID control parameter, feedforward gain, and differential pressure control gain) and servo valve delay to the MST. The study revealed that the control gain and the delay of servo valve have significant impact on the MST. Li et al. 25 analyzed the stability of the MST under different TVC parameters. The analysis showed that better system stability can be obtained with the no-load TVC control parameters. The influences of mechanical installation error, displacement measurement error, and bias of servo valve on the MST were considered in a study by Zhang 26 . The study shows the MST is only related to the bias of servo valve. And the CSI effects increases with the bias of servo valve. Yan studied the influence of the mass of exciter on the MST and found that the CSI effects increase with the mass of exciter 27 . The influence of the stiffness of hinge on the MST is investigated by Xie 28 . The investigation suggests that the stiffness hinge should be increased as much as possible. Maoult et al. 29 adopted finite element modeling to investigate the phenomenon of structural frequency reduction in large-scale shaking table experiments. The investigation demonstrated that the deformation of shaking table is the main reason for the CSI effects. In other words, the deformation of shaking table is one of the factors that affect the MST. Wang 30 and Li 31 studied the CSI effect between dual shaking tables and structure. The study revealed that the coupling effect between the two shaking tables have significant impact on the MST at the frequency of structure and its surrounding frequency band.
Additionally, Wang et al. 32 analyzed the CSI effects on the flow requirement model of servo valve, the force requirement model of exciter, and the resonance response model of load. The analysis was conducted under different load mass, the damping ratio of load, and the frequency of load conditions. The analysis shows the CSI must be considered in designing of the actuator force of actuator, and the flow requirement of servo valve can be achieved without considering the CSI when the flow requirement in the low-frequency band is satisfied.  34 investigated the influence of different test structures on the control performance of shaking table, and the investigation showed that the control performance is greatly affected by the eccentricity of test structure. Therefore, it is necessary to study the CSI effects between shaking table and eccentric load. Moreover, the eccentric degree of load and the moment of inertia of load are important characteristics that have not been analyzed in previous studies. Furthermore, no effect assessment has been proposed to evaluate the CSI effects, and thus, the existing research needs to be further expanded.
In this study, an analytical transfer function matrix is established to analyze the CSI effects between twin-axes shaking table and eccentric load. Based on the transfer function matrix, an in-depth investigation is conducted under different MR, IR, and ER conditions. The sensitivities of the MR, the IR, and the ER to the transfer function matrix, as well as the influence frequency range and the influence trend of the CSI effects, are found in the investigation. Furthermore, a novelty effect assessment is proposed to appraise the CSI effects.

Analytical modeling
The schematic diagram of the twin-axes shaking table and eccentric load is presented in Fig. 2. It can be seen from Fig. 2 that the shaking table is driven by two exciters in horizontal direction, and a centrosymmetric load is loaded on the platform eccentrically. It is worth noting that the ingenious arrangement can simulate the condition that an eccentric load is loaded on the platform to a great extent. Meanwhile, the analytical modeling process of the transfer function matrix is greatly simplified.
The transfer function matrix is established according to the modular method presented in Fig. 3. The method divides the shaking table and eccentric load into three main sub-models, including the dynamic model, the hydraulic-driven model, and the TVC model, which are shown in Fig. 3a. It should be noted that the components and the physical characteristics of the three sub-models are comprehensively considered in the modeling process. Detailed explanations of symbols are given in the following description of the modeling process, and the values of the parameters are listed in Table S1 in the Supplementary Materials.
Dynamic system modeling. The dynamic model is the motion mechanism part of the shaking table and eccentric load. Without regard to the stiffness and the damping between the platform and the connecting rod, the dynamic model of the shaking table and eccentric load is shown in Fig. 3b-d. The detailed explanation of symbols is given as follows. M T is the mass of the platform, M L is the mass of the load, E 1 is exciter 1 of the shaking table, E 2 is exciter 2 of the shaking table, O 1 is the center of gravity of the platform, O 2 is the center of gravity of load, F 1 is the exciter force of exciter 1, F 2 is the exciter force of exciter 2, x 1 is the displacement of exciter 1, x 2 is the displacement of exciter 2, I T is the moment of inertia of the platform relative to its own centroid axis, I L is the moment of inertia of the load relative to its own centroid axis, M E is the equivalent mass of the platform and load, I E is the equivalent moment of inertia of the platform and load relative to its own centroid axis, O E is the center of gravity of equivalent mass, x is the displacement of the equivalent mass, and φ is the motion angle of equivalent moment of inertia, a is the distance from the center of gravity of the equivalent mass to the center of gravity of shaking table, and l is the distance from exciter to the center of gravity of shaking table.  where I T +I TA is the moment of inertia of the shaking table relative to the x-axis in Fig. 3d, I L +I LA is the moment of inertia of the load relative to the x-axis in Fig. 3d. The calculation method of the moment of inertias is given detailly in the Supplementary Materials. Based on Eq. (1), the Newton's second law is adopted to establish the dynamic model of the shaking table and eccentric load. Then the dynamic model is Hydraulic-driven system modeling. The hydraulic-driven system is the driven part of the shaking table and eccentric load. The servo valve and the hydraulic cylinder are the two core components of hydraulic-driven system, the schematic diagram of which is demonstrated in Fig. 3e.
Servo valve. Generally, servo valve can be regarded as a second-order oscillation link 35 , and the transfer function of servo valve is where k q0 is the flow gain of servo valve, n q is the frequency of servo valve, and D q is the damping ratio of servo valve.
Hydraulic cylinder. Combined with servo valve and hydraulic cylinder, the hydraulic-driven system can be expressed by a series of continuity equations 36 . The Laplace transform is applied to the continuity equations, and the converted form can be described as follows  www.nature.com/scientificreports/ where Q L is the flow of servo valve, G q k q0 is the transfer function of servo valve, u is the control error signal, p L is the load pressure, A P is the effective area of cylinder, x T is the displacement of the shaking table, V is the total capacity of two hydraulic cylinder chambers, β is effective bulk modulus, and M TL is the total mass of load, platform and exciter piston.
TVC system modeling. The TVC model is the control part of the shaking table and eccentric load. The TVC model consists of feedback loop, feedforward loop, and generator of the TVC, the schematic diagram of which is presented in Fig. 3a.
According to the TVC system modeling process presented in Ref. 32 , the control error signal is where u is control error signal, u 0 is control signal, G 3 is the generator and the feedforward of the TVC, G 4 is the feedback of the TVC, x T is the displacement of shaking table, and G a is the transfer function of sensors. The analytical expression of G a is where n a is the frequency of sensors and D a is the damping ratio of sensors.
Transfer function matrix modeling. The analytical transfer function matrix is established based on the three sub-models. Assuming the parameters of exciter 1 and 2 are the same, the transfer function matrix can be described as where u 1 and u 2 are the control error signals of two exciters, G q k q0 is the transfer function of servo valve, and the expression of G 2 is Substituting Eqs. (5), (6) and (8) into Eq. (7), the analytical transfer function matrix can be obtained as follows: Converting Eq.

Analysis of the CSI effects
Based on the transfer function matrix, an in-depth analysis is conducted to find the sensitivity of the MR, IR, and ER to the transfer function matrix, and the influence trend and degree of the CSI effects on the transfer function matrix. The analysis is carried out under different MR, IR, and ER conditions. According to the Tables S2 and S3 in the Supplementary Materials, the different MR, IR, and ER conditions are presented in Table 1. In Table 1, the MR is denoted by MR = M L /M T , the IR is denoted by IR = I L /I T , and the ER is denoted by ER = a/l ( l is 1.2 m). For comparing and analyzing the CSI effects on the transfer function matrix, the condition that the shaking table with no load is defined as a reference condition. In the analysis, the absolute value of the amplitude frequency characteristics of H 11 and H 22 is taken as a quantization index. If the quantization index exceeds ± 3.00 dB, it means that the shaking table will not work in its working frequency range. Figure 4 presents the CSI effect on the transfer function matrix under different MR, IR, and ER conditions. Figure 4a shows that at 35.60 Hz the value of H 11 is − 3.00 dB when MR = 0.50, at 27.90 Hz the value of H 11 is − 3.00 dB when MR = 1.0, and at 23.60 Hz the value of H 11 is − 3.00 dB when MR = 1.50. At the same time, with the changes of the MR (from 0.5 to 1.5), the value of H 22 remains to be − 3.00 dB at 48.60 Hz. Based on the above data, it can be obtained that at the resonance peak of oil column frequency and its surrounding frequency band, the influence of the CSI on H 11 increases with the MR, and the MR has no obvious effect on H 22 . Besides, the frequency of the resonance peak of oil column of H 11 decreases with the MR, and the value of the resonance peak of oil column of H 11 increases with the MR. It can be observed from Fig. 4b that at 10.00 Hz, the value of H 12 and H 21 is − 49.40 dB under the reference condition, but it is − 32.10 dB when MR = 0.5, − 23.10 dB when MR = 1.0, and − 18.00 dB when MR = 1.5. According to these collected data, it can be concluded that with the increase of the MR, the coupling between two exciters increases rapidly. The coupling is amplified at least 31.40 dB (about 37.15 times).     Fig. 4, the MR, ER, and IR have no evident effect on the phase diagram of the transfer function matrix (the quantization index does not exceed ± 60°), and thus, the phase diagram is not discussed in this paper. Besides, the role of the ER can be identified as follows. The ER exerts effects on the transfer function matrix at the resonance peak of oil column frequency and its surrounding frequency band.

Assessment of the CSI effects
The above "Analysis of the CSI effects" proves that the CSI exerts different degrees of effects on the transfer function matrix. However, the above analysis needs to be further expanded to appraise the CSI effect in practical experiments. To appraise whether the CSI effects can be ignored in shaking table test, a novelty effect assessment is proposed. Two steps are involved in proposing the assessment. The first step is the deduction of the assessment, and the second step is the visualization expression of the assessment.
Deduction of the effect assessment. The "Analysis of the CSI effects" proves that the MR, IR, and ER have influence on the H 11 and H 22 . In the light of Eq. (1) and the parallel axis theorem, the MR, IR, and ER are related to the mass and the inertia moment of the system. The relationship indicates that the CSI effects are related to two physical quantities, the moment of inertia and the mass of the system. In the deduction of the effect assessment, the ratio of the moment of inertia of shaking table to the moment of inertia of the system (IRS) and the MR are adopted to represent the two physical quantities. It is assumed that the assessment can be expressed as follows: where IRS is the ratio of the I T to the I E , and the expression of IRS is (16)  (22) that the CSI effects are related to the MR and ER. Based on the analysis, a visualization expression of the assessment can be obtained. The concept for obtaining the visualization expression is presented as follows. Firstly, to appraise whether the CSI effects can be ignored, a judgment standard needs to be proposed. The standard is that if the amplitude characteristics of H 11 and H 22 exceed the range of ± 3.00 dB or the effective frequency band of H 11 and H 22 decreases to 70% of the frequency band of the reference condition, the CSI effects cannot be ignored. Secondly, the values of the MR and ER are determined according to the standard. In the determination process, the range of the MR is investigated under the assumption of ER = 0, and then the ER is obtained under specific value of MR.
According to the concept, the characteristics of the H 11 and H 22 under different MR are determined. The result of the determination is presented in Fig. 5. It can be seen from Fig. 5  Based on the conclusion that the MR ≤ 0.75, the specific values of MR, − 0.75, 0.60, 0.45, and 0.30, are adopted to obtain the values of the ER. Then the coordinate points ( MR , ER ) are drawn in rectangular coordinate system, and a curve can be obtained by connecting the points (MR, ER). The curve is the visualization expression of the assessment of which is shown Fig. 6. To illustrate the specific meaning of the coordinate points, the point (0.45, 0.0875) is taken as an example. The point (0.45, 0.0875) indicates that when MR = 0.45, the CSI effects can be ignored if ER ≤ 0.0875. If a point (MR, ER) is above the curve, the CSI cannot be ignored, and if a point (MR, ER) is below the curve, then the CSI can be ignored.

Conclusions
In this study, an analytical transfer function matrix is established to analyze the CSI effects between the twinaxes shaking table and eccentric load. Based on the transfer function matrix, a comprehensive investigation is conducted under different MR , IR , and ER conditions. The sensitivities of the MR , IR , and ER to the transfer function matrix, the influence frequency range and the influence trend of the CSI effects are obtained. Furthermore, a novelty effect assessment is proposed to appraise whether the CSI effects can be ignored in shaking table test. The most important conclusions can be drawn as follows.  www.nature.com/scientificreports/ (1) The ER exerts effects on the transfer function matrix at the resonance peak of oil column frequency and its surrounding frequency band. (2) The transfer function matrix is the most sensitive to the changes of the MR, followed by the ER and then the IR. The IR has slight influence on the transfer function matrix. Additionally, the CSI exerts different influence trend and the influence degree on the transfer function matrix under different MR, ER, and IR conditions. (3) The CSI leads to a significant increase in the exciter coupling. The coupling between two exciters is amplified at least 37.15 times. (4) The novelty effect assessment has a wide application prospect and practical value in appraising whether the CSI effects can be ignored in shaking table test. (5) It is worth expanding the assessment to appraise the CSI effect between shaking table and flexible structure in the future.

Data availability
All data is available in the main text or the Supplementary Information. The data are available from the corresponding author upon reasonable request.